Find probability distribution - choosing points from a circle

Question

From a disk R we choose a point. Let X denote distance between chosen point and circle's centre. Find distribution of a variable $X^2$. I have no idea what should i do about that whatsoever so i would appreciate detailed explanation.

Answer 1

Hint/Outline:

1. You mean to say pick a point randomly (uniformly) inside the circle. Then $X$ represents the distance from the origin to that point. I would approach this using the cdf, meaning compute $P(X<x)$.

2. Once you have found the distribution of $X$, use your preferred method to find the distribution of $Y=X^2$.

It might help to draw some pictures.

Answer 2

The point $P$ is chosen uniformly in the disk with centre $O$. Thus the area of the disk with centre $O$, and going through $P$, is uniformly distributed on the interval $[0,\pi R^2]$. But this area is $\pi X^2$, so $X^2$ is uniformly distributed on $[0,R^2]$.